Abstract
An efficient parallel multiscale numerical algorithm is proposed for a parabolic equation with rapidly oscillating coefficients representing heat conduction in composite material with periodic configuration. Instead of following the classical multiscale asymptotic expansion method, the Fourier transform in time is first applied to obtain a set of complex-valued elliptic problems in frequency domain. The multiscale asymptotic analysis is presented and multiscale asymptotic solutions are obtained in frequency domain which can be solved in parallel essentially without data communications. The inverse Fourier transform will then recover the approximate solution in time domain. Convergence result is established. Finally, a novel parallel multiscale FEM algorithm is proposed and some numerical examples are reported.
Highlights
Consider the parabolic equation with rapidly oscillating coefficients as follows: ∂uε (x, ∂t t) − ∂ ∂xi (aij ( x ε ) ∂uε (x, t) ) ∂xj = f (x, t)(x, t) ∈ Ω × (0, T), (1) uε (x, t) = 0, (x, t) ∈ ∂Ω × (0, T), uε (x, 0) = u (x), x ∈ Ω, where Ω ∈ Rn is a bounded smooth domain or a bounded polygonal convex domain with a periodic microstructure and (0, T) denotes the time domain
The unknown is the temperature increment uε(x, t). f(x, t) and u(x) are some given functions which denote heat source and initial temperature, respectively. ε > 0 is a small parameter and it represents the relative size of a periodic cell
Several efficient numerical methods have been proposed and analyzed; for instance, see [7,8,9]. From these literatures, we found that the homogenized equation happens to be Mathematical Problems in Engineering a system of the same type with constant coefficients
Summary
Consider the parabolic equation with rapidly oscillating coefficients as follows:. (x, t) ∈ Ω × (0, T) , (1) uε (x, t) = 0, (x, t) ∈ ∂Ω × (0, T) , uε (x, 0) = u (x) , x ∈ Ω, where Ω ∈ Rn is a bounded smooth domain or a bounded polygonal convex domain with a periodic microstructure and (0, T) denotes the time domain. Several efficient numerical methods have been proposed and analyzed; for instance, see [7,8,9] From these literatures , we found that the homogenized equation happens to be Mathematical Problems in Engineering a system of the same type with constant coefficients. We take the Fourier transform with respect to the time variable in the parabolic equation and present the multiscale asymptotic expansions of the solutions for the resulting equations in space-frequency domain at specific frequency points of interest, which are independent and may be done in parallel. The approximate solution of parabolic equation in space-time domain is retrieved by the discrete inverse Fourier transform. All complex-valued functions are assumed to have values in the complex field C and standard notations for function spaces and their associated norms will be used in this paper
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